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arXiv:2610.37861

The Geometry of Quantum Teleportation in Phase Space

Jasel Berra-Montiel, Alberto Molgado, Mar Sánchez-Córdova

quant-phmath-phSubmitted Oct 27, 2026• Announced Oct 29, 2026• 0 citations
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Abstract

We formulate the standard quantum teleportation protocol within the framework of deformation quantization, employing the Stratonovich--Weyl (SW) correspondence and the associated star product on the coadjoint orbits of $SU(2)$. Each qubit is represented by a spherical phase space $S^2$ endowed with the Kirillov--Kostant--Souriau (KKS) symplectic structure. Within this setting, we explicitly construct the phase space representations of the entangled Bell resource, the controlled-NOT and Hadamard operators, the projective measurements, and the conditional Pauli corrections. We demonstrate how the quantum information initially encoded in the Wigner function for the input qubit is redistributed through bipartite and tripartite phase space correlations. Upon consideration of a conditional unitary transformation, the measurement outcome produces a Wigner function on the receiver's coadjoint orbit that is related to the input by an outcome-dependent rotation, which can subsequently be inverted through the implementation of an appropriate Pauli correction. The covariance of the SW correspondence provides a geometric interpretation of this process, in which quantum information is redistributed and reconstructed without transferring the underlying symplectic geometry. Our formulation can also be extended to study quantum teleportation in more general phase space geometries equipped with suitable star products.

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We formulate the standard quantum teleportation protocol within the framework of deformation quantization, employing the Stratonovich--Weyl (SW) correspondence and the associated star product on the coadjoint orbits of SU(2)SU(2)SU(2). Each qubit is represented by a spherical phase space S2S^2S2 endowed with the Kirillov--Kostant--Souriau (KKS) symplectic structure. Within this setting, we explicitly construct the phase space representations of the entangled Bell resource, the controlled-NOT and Hadamard operators, the projective measurements, and the conditional Pauli corrections. We demonstrate how the quantum information initially encoded in the Wigner function for the input qubit is redistributed through bipartite and tripartite phase space correlations. Upon consideration of a conditional unitary transformation, the measurement outcome produces a Wigner function on the receiver's coadjoint orbit that is related to the input by an outcome-dependent rotation, which can subsequently be inverted through the implementation of an appropriate Pauli correction. The covariance of the SW correspondence provides a geometric interpretation of this process, in which quantum information is redistributed and reconstructed without transferring the underlying symplectic geometry. Our formulation can also be extended to study quantum teleportation in more general phase space geometries equipped with suitable star products.
Description This laboratory work presents a practical and theoretical study of Pauli matrices (X, Y, Z) and their fundamental role in quantum teleportation implemented on IBM Quantum hardware. The work explores the mathematical structure of single-qubit operations in a two-dimensional Hilbert space and their geometric interpretation as rotations on the Bloch sphere. A Bell state is prepared using Hadamard and CNOT gates, forming the entangled resource required for teleportation. The protocol is analyzed in matrix form, demonstrating how measurement outcomes correspond to Pauli correction operators (I, X, Z, XZ). The study emphasizes that quantum teleportation represents controlled amplitude and phase correction based on classical measurement results rather than physical transfer of matter. Special attention is given to the applied implementation on IBM Quantum, where Pauli operators are realized through physical microwave pulses corresponding to exponentials of SU(2) generators. While the work operates at the level of quantum circuits, it highlights the underlying algebraic structure of the Lie algebra su(2), which governs qubit rotations and quantum state evolution. The report integrates mathematical formalism, circuit-level realization, and physical interpretation, providing a coherent understanding of the role of Pauli operators in quantum information processing.
Quantum teleportation is reinterpreted as a natural geometric process within Unified Relational Projection Dynamics (URPD).Instead of treating teleportation as a special quantum protocol, this framework views quantum states as relational frames in the projective space CPN−1\mathbb{CP}^{N-1}CPN−1.Entanglement establishes a high-overlap shortcut between distant frames, and a Bell measurement corresponds to a discrete transition in this geometric structure.Classical communication is required not as an add-on but as a direct consequence of causal constraints encoded in URPD.The no-cloning theorem, perfect fidelity, and the structure of the teleportation protocol all emerge from the geometry of relational frames.This work shows that teleportation is an inevitable feature of the informational geometry underlying URPD, providing the first unified explanation of quantum state transfer, the Born rule, and relativistic causality within a single theoretical framework.
FINDING: Quantum teleportation relies on shared entanglement and classical communication, not instantaneous matter transfer; mathematical core is Bell state measurement and unitary transformation. | MATH: Bell state basis: ∣Φ±⟩=12(∣00⟩±∣11⟩)|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)∣Φ±⟩=2​1​(∣00⟩±∣11⟩), ∣Ψ±⟩=12(∣01⟩±∣10⟩)|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)∣Ψ±⟩=2​1​(∣01⟩±∣10⟩). Teleportation protocol: Alice performs Bell measurement on her qubit and entangled pair, sends 2 classical bits to Bob, who applies Pauli correction I,X,Z,XZI, X, Z, XZI,X,Z,XZ based on outcome. Fidelity = ∣⟨ψ∣ρout∣ψ⟩∣2|\langle \psi | \rho_{\text{out}} | \psi \rangle|^2∣⟨ψ∣ρout​∣ψ⟩∣2. No faster-than-light signaling: classical bits limit speed to ≤ c. | CONNECTION: No direct geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) appear. However, Bell states are maximally entangled, corresponding to the four vertices of a regular tetrahedron in the Bloch sphere representation (SO(3) symmetry). The Pauli matrices X,Y,ZX, Y, ZX,Y,Z generate the Clifford group, which is the sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum teleportation is a protocol for transferring an unknown quantum state via shared entanglement and classical communication — not physical matter transport. The core math is the Bell basis projection and the Pauli correction operators. MATH: - State to teleport: ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle∣ψ⟩=α∣0⟩+β∣1⟩, with ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1∣α∣2+∣β∣2=1. - Shared entangled pair: Bell state ∣Φ+⟩=12(∣00⟩+∣11⟩)|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)∣Φ+⟩=2​1​(∣00⟩+∣11⟩). - Total 3-qubit state: ∣ψ⟩⊗∣Φ+⟩|\psi\rangle \otimes |\Phi^+\rangle∣ψ⟩⊗∣Φ+⟩. - Alice performs Bell measurement — projecting onto one of 4 Bell states: ∣Φ±⟩=12(∣00⟩±∣11⟩)|\Phi^\pm\rangle = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle)∣Φ±⟩=2​1​(∣00⟩±∣11⟩), ∣Ψ±⟩=12(∣01⟩±∣10⟩)|\Psi^\pm\rangle = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle)∣Ψ±⟩=2​1​(∣01⟩±∣10⟩). - Outcome determines Pauli correction on Bob's qubit: I,X,Z,XZI, X, Z, XZI,X,Z,XZ (identity, bit-flip, phase-flip, both). - No-cloning theorem: the original state is destroyed — information is transferred, not copied. - Classical communicati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of SU(2)SU(2)SU(2) and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on S2S^2S2 and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.
The tomographic representation of quantum fields within the deformation quantization formalism is constructed. By employing the Wigner functional we obtain the symplectic tomogram associated with quantum fields. In addition, the tomographic version of the Wigner map allows us to compute the symbols corresponding to field operators. Finally, the functional integral representation of the tomographic star product is determined. Some possible applications of the formalism to loop quantum cosmology and loop quantum gravity are briefly discussed.